14 Geometric Progression Annuities

14.1 Geometric Progression Annuity Formulae

An annuity with payments that are a product of the preceding payment and a constant factor is known as a geometric progression annuity.

Consider an annuity-due that pays $1$ at the beginning of the first unit of time, $(1+g)$ at the beginning of the second unit of time, $(1+g)^2$ at the beginning of the third year, and so on until the final payment of $(1+g)^{n-1}$ at the beginning of the $n$-th year.

If $g$ is positive, then the payments are increasing, and their rate of growth is $g$. The present value of this annuity is: \[\begin{align} PV_0 &= 1 + (1+g)v + (1+g)^2v^2 + ... + (1+g)^{n-1} v^{n-1} \\ &= 1 + \bigg(\frac{1+g}{1+i}\bigg) + \bigg(\frac{1+g}{1+i}\bigg)^2 + ... + \bigg(\frac{1+g}{1+i}\bigg)^{n-1} \\ &= \ddot a_{\overline{n|}j }\end{align}\] where $1 + j = \frac{1+i}{1+g}$.

The accumulated value of the annuity is found by accumulating the annuity for $n$ year: \[AV_n = (1+i)^n PV_0 = (1+i)^n \ddot{a}_{\overline{n|}j}\] Note that, to accumulate we must use $i$, but not $j$.

14.1.1 Geometric Annuity-Due, Payable Once per Time Unit

The present value at time $0$ and the accumulated value at time $n$ of an annuity that pays $1$ at the beginning of the first unit of time, $(1+g)$ at the beginning of the second unit of time, $(1+g)^2$ at the beginning of the third year, and so on until the final payment of $(1+g)^{n-1}$ at the beginning of the $n$-th year are:

\[PV_0 = \ddot a_{\overline{n|}j} \hskip7em AV_n = (1+i)^n \ddot a_{\overline{n|}j}\] where $j$ is defined such that \[1+j = \frac{1+i}{1+g}\]

14.1.2 Geometric Annuity-Immediate, Payable Once per Time Unit

The present value at time $0$ and the accumulated value at time $n$ of an annuity that pays $1$ at the end of the first unit of time, $(1+g)$ at the end of the second unit of time, $(1+g)^2$ at the end of the third year, and so on until the final payment of $(1+g)^{n-1}$ at the end of the $n$-th year are: \[PV_0 = \frac{\ddot a_{\overline{n|}j}}{1+i} \hskip7em AV_n= (1+i)^{n-1}\ddot a_{\overline{n|}j} \] where $j$ is defined such that \[1+j = \frac{1+i}{1+g}\]

Example: An annual, 10-year geometric annuity-due makes a payment of $\$100$ now. Each subsequent payment is $7\%$ greater than the preceeding payment. The annual effective rate of interest is $5\%$. Calculate the present value of the annuity-due.

Solution: The value of $j$ is: \[j = \frac{i-g}{1+g} = \frac{0.05-0.07}{1.07} = -0.01869\] The present value is: \[PV_0 = 100\times \ddot a_{\overline{n|}j} = 100\times \frac{1-(1-0.01869)^{-10}}{\frac{-0.01869}{1-0.01869}} = 100 \times 10.9022 = 1,090.22\]

Using the BA II, we have:

0.05 [-] 0.07 [=] [÷] 1.07 [×] 100 [I/Y]
10 [N] 100 [PMT] 0 [FV]
[2nd] [BGN] [2nd] [SET] [2nd] [QUIT]
[CPT] [PV] -> PV = -1,090.22